# Chain rules for canonical state extensions on von Neumann algebras

Colloquium Mathematicae (1993)

- Volume: 64, Issue: 1, page 115-119
- ISSN: 0010-1354

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topCecchini, Carlo, and Petz, Dénes. "Chain rules for canonical state extensions on von Neumann algebras." Colloquium Mathematicae 64.1 (1993): 115-119. <http://eudml.org/doc/210160>.

@article{Cecchini1993,

abstract = {In previous papers we introduced and studied the extension of a state defined on a von Neumann subalgebra to the whole of the von Neumann algebra with respect to a given state. This was done by using the standard form of von Neumann algebras. In the case of the existence of a norm one projection from the algebra to the subalgebra preserving the given state our construction is simply equivalent to taking the composition with the norm one projection. In this paper we study couples of von Neumann subalgebras in connection with the state extension. We establish some results on the ω-conditional expectation and give a necessary and sufficient condition for the chain rule of our state extension to be true.},

author = {Cecchini, Carlo, Petz, Dénes},

journal = {Colloquium Mathematicae},

keywords = {extension of a state; von Neumann subalgebra; standard form of von Neumann algebras; norm one projection; state extension; -conditional expectation; chain rule},

language = {eng},

number = {1},

pages = {115-119},

title = {Chain rules for canonical state extensions on von Neumann algebras},

url = {http://eudml.org/doc/210160},

volume = {64},

year = {1993},

}

TY - JOUR

AU - Cecchini, Carlo

AU - Petz, Dénes

TI - Chain rules for canonical state extensions on von Neumann algebras

JO - Colloquium Mathematicae

PY - 1993

VL - 64

IS - 1

SP - 115

EP - 119

AB - In previous papers we introduced and studied the extension of a state defined on a von Neumann subalgebra to the whole of the von Neumann algebra with respect to a given state. This was done by using the standard form of von Neumann algebras. In the case of the existence of a norm one projection from the algebra to the subalgebra preserving the given state our construction is simply equivalent to taking the composition with the norm one projection. In this paper we study couples of von Neumann subalgebras in connection with the state extension. We establish some results on the ω-conditional expectation and give a necessary and sufficient condition for the chain rule of our state extension to be true.

LA - eng

KW - extension of a state; von Neumann subalgebra; standard form of von Neumann algebras; norm one projection; state extension; -conditional expectation; chain rule

UR - http://eudml.org/doc/210160

ER -

## References

top- [1] L. Accardi and C. Cecchini, Conditional expectations in von Neumann algebras and a theorem of Takesaki, J. Funct. Anal. 45 (1982), 245-273. Zbl0483.46043
- [2] C. Cecchini and D. Petz, State extension and a Radon-Nikodym theorem for conditional expectations on von Neumann algebras, Pacific J. Math. 138 (1989), 9-24. Zbl0695.46024
- [3] C. Cecchini and D. Petz, Classes of conditional expectations over von Neumann algebras, J. Funct. Anal. 92 (1990), 8-29.
- [4] A. Connes, Sur le théorème de Radon-Nikodym pour les poids normaux fidèles semifinis, Bull. Sci. Math. Sect. II 97 (1973), 253-258.
- [5] A. Connes, On a spatial theory of von Neumann algebras, J. Funct. Anal. 35 (1980), 153-164. Zbl0443.46042
- [6] D. Petz, Sufficient subalgebras and the relative entropy of states of a von Neumann algebra, Comm. Math. Phys. 105 (1986), 123-131. Zbl0597.46067

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